A generalized Boolean function generator for complementary sequences (original) (raw)

A Generalized Boolean Functions Generator for Complementary Sequences

We present a general algorithm for generating arbitrary standard complementary pairs of sequences (including binary, polyphase, M-PSK and QAM) of length íµí¿ íµí±µ using Boolean functions. The algorithm follows our earlier paraunitary algorithm, but does not require matrix multiplications. The algorithm can be easily and efficiently implemented in hardware. As a special case, it reduces to the non-recursive (direct) algorithm for generating binary sequences given by Golay, to the algorithm for generating M-PSK sequences given by Davis and Jedwab (and later improved by Paterson) and to all published algorithms for generating QAM sequences. The Boolean index form of the algorithm can be directly generalized to complementary sets.

Universal Generator for Complementary Pairs of Sequences Based on Boolean Functions

We present a general algorithm for generating arbitrary standard complementary pairs of sequences (including binary, polyphase, M-PSK and QAM) of length 2^N using Boolean functions. The algorithm follows our earlier paraunitary algorithm, but does not require matrix multiplications. The algorithm can be easily and efficiently implemented in hardware. As a special case, it reduces to the non-recursive (direct) algorithm for generating binary sequences given by Golay, to the algorithm for generating M-PSK sequences given by Davis and Jedwab (and later improved by Paterson) and to all published algorithms for generating QAM sequences. However the algorithm does not solve the problem of sequence uniqueness (except for the trivial M-PSK case), which must be treated separately for each QAM constellation.

A Boolean Generator for QAM and Other Complex Complementary Sequences of Length 2^K

A Boolean generator for a broad set of standard pairs of complex valued complementary sequences of length 2^K is proposed. Binary, M-PSK and rectangular or hexagonal QAM sequences can be generated. The Boolean generator, which uses scalar multiplications, is derived from our paraunitary algorithm from 2013 based on matrix multiplication. Both algorithms are based on unitary matrices which are mostly unimodular except a few that are called QAM unitary matrices (Qum). Also, in contrast to previous Boolean QAM algorithms proposed by Li in 2010 and Zilong et al. in 2013, which have an additive form, our algorithm has a multiplicative form. Any element of the sequence can be efficiently generated from the outputs of a binary counter. Generalized Case I, II and III sequences given by Li are identical to those generated by our 1-Qum Boolean generator. The generalized Cases IV and V given by Zilong correspond to a special case of our 2-Qum generator. However, the general 2-Qum algorithm generates 9 times more sequences for 64-QAM constellations, while for 4096-QAM constellations it generates 942 times more sequences. Greatest common divisor of Gaussian integers plays a key role in ensuring that the generator generates a complete set of unique sequences and, also, in deriving their enumeration formula.

Paraunitary Construction-Correlation of QAM Complementary Sequence Pairs

A unique decomposition of arbitrary pairs of complementary sequences (including standard binary, polyphase and QAM sequences as well as non-standard sequences and kernels) based on paraunitary matrices is presented. This decomposition allows us to describe the internal structure of any sequence pair of length L using basic paraunitary matrices defined by an ordered set of L complex coefficients named the omega vector. When the omega vector is sparse, the canonic form is compact and leads to an efficient implementation of a generator/correlator. We show that sequences generated by the standard algorithm have the sparsest known omega vector (log_2⁡L non-zero elements out of L) and, thus, the most efficient generator/correlator. The equivalence of paraunitary matrices and Z transforms of complementary sequences allows us to apply the rich results from the theory of perfect reconstruction filter-banks to the field of sequence design. We introduce a new generator/correlator algorithm for sequences in standard and non-standard QAM constellations that is based on this equivalence. Both rectangular and hexagonal constellations are considered and the cardinality of the generated set of unique complementary sequences is either determined or estimated for a number of important cases. We show, in the case of the standard 16-QAM constellation, that the paraunitary algorithm generates the same number of sequences as the published algorithms based on generalized Boolean functions. In the case of 64-QAM, the proposed algorithm generates more sequences than known algorithms. We introduce an algorithm for generating 256-QAM sequences and derive a tight upper bound on the number of generated sequences.